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Lz#ykC|M0&6&FPLs literal 0 HcmV?d00001 diff --git a/docs/final_notecard.tex b/docs/final_notecard.tex new file mode 100644 index 0000000..c4ae60e --- /dev/null +++ b/docs/final_notecard.tex @@ -0,0 +1,352 @@ +\documentclass[sigconf,authorversion,nonacm]{acmart} + +\usepackage{bbold} +\usepackage{geometry} +\geometry{margin=0.3in} + +\nonstopmode + +\begin{document} +\title{CSCI577 Final} + +\maketitle + +\section{Classification} +\paragraph{Definition} +Constructing a method of classifying new instances using information in a training set + +\begin{itemize} + \item Naive Bayes (Conditional Probabilities) + \item Decision Trees + \item Logistic Regression + \item Neural Networks +\end{itemize} + +\subsection{TDIDT} +\paragraph{Definition} +Top-Down Induction of Decision Trees + +\subsection{Adequacy} +\paragraph{Definition} +No two instances with the same values of all the attributes may belong to different classes. +Naive bayes can still be used when this doesn't obtain, as it will still be able to obtain the probabilities of each class. +kNN can still be used, as long as then multiple datapoints in the same location in euclidean space would still function as expected. + +\paragraph{Application} +TDIDT algrorithm is negatively affected + +\paragraph{Algorithm} +Until no more splitting is possible: +\begin{itemize} + \item IF all the instances in the training set belong to the same class THEN return the value of the class + \item ELSE + \begin{enumerate} + \item (a) Select an attribute A to split on + \item (b) Sort the instances in the training set into subsets, one for each value of attribute A + \item (c) Return a tree with one branch for each non-empty subset + \begin{itemize} + \item Each branch having a descendant subtree or a class value produced by applying the algorithm recursively + \end{itemize} + \end{enumerate} +\end{itemize} + +\subsection{Overfitting} +Understand the concept of overfitting and be able to tell how you would know that a classification system overfit +\paragraph{Definition} +If classifier generates a decision tree (or other mechanism) too well adapted to the training set +Performs well on training set, not well on other data. +Some overfitting inevitable. + +Remedy: +\begin{itemize} + \item Adjust a decision tree while it is being generated: Pre-pruning + \item Modify the tree after creation: Post-pruning +\end{itemize} + +\subsubsection{Clashes} +Two (or more) instances of a training set have identical attribute values, but different classification. +Especially a problem for TDIDT's 'Adequacy condition'. + +\paragraph{Stems from} +\begin{itemize} + \item Classification incorrectly recorded + \item Recorded attributes insufficient - Would need more attributes, normally impossible +\end{itemize} + +\paragraph{Solutions} +\begin{itemize} + \item Discard the branch to the clashing node from the node above + \item Of clashing instances, assume the majority label +\end{itemize} + +\subsubsection{Prepruning} +When pre-pruning, may reduce accuracy on training set but may be better on test set (and subsequent) data than unpruned classifier. +\begin{enumerate} + \item test whether a termination condition applies. + \begin{itemize} + \item If so, current subset is treated as a 'clash set' + \item Resolve by 'delete branch,' 'majority voting,' etc. + \end{itemize} + \item two methods methods + \begin{itemize} + \item Size Cuttoff – prune of subset has fewer than X instances + \item Maximum depth: prune if length of branch exceed Y + \end{itemize} +\end{enumerate} + +\subsubsection{PostPruning} +\begin{enumerate} + \item look for a non-leaf nodes that have descendants of length 1. + \item In this tree, only node G and D are candidates for pruning (consolidation). +\end{enumerate} + + +\subsection{Discretizing} + +\subsubsection{Equal Width Intervals} +\subsubsection{Pseudo Attributes} +\subsubsection{Processing Sorted Instance Table} +\subsubsection{ChiMerge} + +\paragraph{Rationalization} +Initially, each distinct value of a numerical attribute $A$ is considered to be one interval. +$\chi^{2}$ tests are performed for every pair of adjacent intervals. +Adjacent intervals with the least $\chi^{2}$ values are merged together, because $\chi^{2}$ low values for a pair indicates similar class distributions. +This merging process proceeds recursively until a predefined stopping criterion is met. +For two adjacent intervals, if $\chi^{2}$ test concludes that the class is independent intervals should be merged. +If $\chi^{2}$ test concludes that they are not independent, i.e. the difference in relative class frequency is statistically significant, the two intervals should remain separate. + +\paragraph{Calculation} +To calculate expected value for any combination of row and class: +\begin{enumerate} + \item Take the product of the corresponding row sum and column sum + \item Divided by the grand total of the observed values for the two rows. +\end{enumerate} + +Then: +\begin{enumerate} + \item Using observed and expected values, calculate, for each of the cells: + \begin{math} + \frac{(O - E)^{2}}{E} + \end{math} + \item Sum each cell's $\chi^{2}$ +\end{enumerate} +When exceeds $\chi^{2}$ threshold, hypothesis is rejected. +Small value for supports hypothesis. +Important adjustment, when $E < 0.5$ replace it with $0.5$. +\begin{enumerate} + \item Select the smallest value + \item Compare it to the threshold + \item If it falls below the threshold, merge it with the row immediately below it + \item recalculate $\chi^{2}$, Only need to do this for rows adjacent to the recently merged one. +\end{enumerate} + +Large numbers of intervals does little to solve the problem of discretization. +Just one interval cannot contribute to a decision making process. +Modify significance level hypothesis of independence must pass, triggering interval merge. +Set a minimum and a maximum number of intervals + +\subsection{Entropy} +\paragraph{Definition} +Entropy is the measure of the presence of there being more than one possible classification. +Used for splitting attributes in decision trees +Entropy minimizes the complexity, number of branches, in the decision tree. +No guarantee that using entropy will always lead to a small decision Tree +Used for feature reduction: Calculate the value of information gain for each attribute in the original dataset. Discard all attributes that do not meet a specified criterion. +Pass the revised dataset to the preferred classification algorithm + +Entropy has bias towards selecting attributes with a large number of values +\paragraph{Calculation} +To decide if you split on an attribute: +\begin{enumerate} + \item find the entropy of the data in each of the branches after the split + \item then take the average of those and use it to find information gain. + \item The attribute split with the highest information gain (lowest entropy) is selected. +\end{enumerate} + +\begin{itemize} + \item Entropy is always positive or zero + \item Entropy is zero when $p_{i} = 1$, aka when all instances have the same class + \item Entropy is at its max value for the # of classes when all classes are evenly distributed +\end{itemize} + +If there are classes, we can denote the proportion of instances with classification $i$ by $p_{i}$ for $i = 1 to K$. +$p_{i} = \frac{\text{instances of class} i}{\text{total number of instances}}$ + +\begin{displaymath} + \text{Entropy} = E = -\sum_{i=1}^{K} p_{i} log_{2} p_{i} +\end{displaymath} + +where $K =$ non-empty classes and $p_{i} = \frac{\lvert i \rvert}{N}$, instances in class $i$ over total number of instances $N$. + + +\subsection{GINI} + +\paragraph{Calculation} +\begin{enumerate} + \item For each non-empty column, form the sum of the squares of the values in the body of the table and divide by the column sum. + \item Add the values obtained for all the columns and divide by N (the number of instances). + \item Subtract the total from 1. +\end{enumerate} + +\subsection{Information Gain} +\paragraph{Definition} +The difference between the entropy before and after splitting on a given attribute in a decision tree. +Maximizing information gain is the same as minimizing $E_{new}$. +\paragraph{Calculation} + +\begin{displaymath} + \text{Information Gain} = E_{\text{start}} - E_{\text{new}} +\end{displaymath} + +Starting node: + +\begin{eqnarray} + E_{\text{start}} = -\frac{4}{24}log_{2}\frac{4}{24} \\ \nonumber + -\frac{5}{24}log_{2}\frac{5}{24} \\ \nonumber + -\frac{15}{24}log_{2}\frac{15}{24} +\end{eqnarray} + +After spliting on attribute: + +\begin{eqnarray} + E_{\text{new}} = \frac{8}{24}E_{1}\\ \nonumber + + \frac{8}{24}E_{2}\\ \nonumber + + \frac{8}{24}E_{3}\\ \nonumber +\end{eqnarray} + + +\paragraph{Uses} + + +\section{Clustering} +\paragraph{Definition} +Grouping data into seperate groups. +Use distance metric between two datapoints. +Groups should be distinct from another and composed of items similar to one another, and different from items in other groups. + +\subsection{Naïve Bayes} +\begin{displaymath} + P(c_{i} | v) = P(c_{i}) \prod_{j=1}^{n} P(a_{j} = v_{j} | \text{class} = c_{i}) +\end{displaymath} + +\subsection{Nearest Neighbors} +Mainly used when all attribute values are continuous + +General strategy: +\begin{enumerate} + \item Find the $k$ training instances that are closest to the unseen instance. + \item Take the most commonly occurring classification for these instances. +\end{enumerate} + +\begin{itemize} + \item KMeans + \item DBSCAN +\end{itemize} + + +\section{Sequence Mining} +\textbf{TODO} +\paragraph{Definition} +Finding meaningful, recurring sequences of events + + +\section{Association Rule Analysis} +\paragraph{Definition} +Given a collection of collections (database of transactions of food items), find items with high co-occurance. + +Let $m$ be the number possible items that can be bought. +Let $I$ denote the set of all possible items. +Possible itemsets: $s^{\lvert I \rvert}$ +An itemset $S$ matches a transaction $T$ (itself an itemset) if $S \subset T$. + +\subsection{Support} + +\paragraph{Definition} +$support(S)$: proportion of itemsets matched by $S$. +Proportion of transactions that contain all the items in $S$. +Frequency with which the items in S occur together in the database. + +\paragraph{Calculation} +\begin{displaymath} + \text{support}(S) = \frac{count(S)}{n} +\end{displaymath} +where n is the number of transactions in the database. + + +\subsubsection{Uses} + +\subsection{Confidence} + +\paragraph{Calculation} +Confidence of a rule can be calculated either by +\begin{displaymath} + Confidence(L \rightarrow R) = \frac{count(L \cup R)}{count(L)} +\end{displaymath} +or +\begin{displaymath} + Confidence(L \rightarrow R) = \frac{support(L \cup R)}{support(L)} +\end{displaymath} +Reject rules where +\begin{displaymath} + support < minsup \approx 0.01 = 1\% +\end{displaymath} +Also called a frequent|large|supported itemset. + +Reject rules where +\begin{displaymath} + confidence < minconf \approx 0.8 = 80\% +\end{displaymath} + +\paragraph{Uses} + +\subsection{Lift} +\paragraph{Definition} +Lift measures how many more times the items in and occur together than would be expected if they were statistically independent. +\paragraph{Calculation} + +\begin{eqnarray} + \text{Lift}(L \rightarrow R) + = \frac{\text{count}(L \cup R)}{\text{count}(L) \times \text{support}(R)} \nonumber + \\ + = \frac{\text{support}(L \cup R)}{\text{support}(L) \times \text{support}(R)} \nonumber + \\ + = \frac{\text{confidence}(L \rightarrow R)}{\text{support}(R)} \nonumber + \\ + = \frac{N \times \text{confidence}(L \rightarrow R)}{\text{count}(R)} \nonumber + \\ + = \frac{N \times \text{confidence}(R \rightarrow L)}{\text{count}(R)} \nonumber + \\ + = \text{Lift}(R \rightarrow L) +\end{eqnarray} + +\paragraph{Uses} + +\subsection{Frequent Itemsets} +\begin{enumerate} + \item Find itemsets of size $k$ made from 2 supported itemsets of size $k-1$ + \item For each new itemset: + \begin{enumerate} + \item check if every sub-itemset in it also exists in the supported itemsets of size $k - 1$. + \item If not every sub-itemset does, then prune it + \begin{enumerate} + \item Now with the final candidates, determine if they have mininum support + \item To determine association rules, find which itemsets have at least minimum confidence +\end{enumerate} + + +\subsection{Rules Possible} + +\begin{displaymath} + \sideset{_k}{_i}C +\end{displaymath} + +or + +\begin{displaymath} + 2^{k} - 2 +\end{displaymath} + + +\end{document} +\endinput